Clarke's tangent cone, Bouligand's tangent cone, and set regularity

Question

For a set $C$ (which may not be convex) and a point $x\in C$:

• Bouligand's Tangent cone is defined as $$ T(C,x) = \left\{v : \lim_{\theta\to 0_+} \inf \frac{d(x+\theta v, C)}{\theta} = 0\right\} $$ and where $d(x,C) = \min_{y\in C} \|x-y\|$ the distance from a point to a set.
• Clarke's tangent cone is $$ T_C(C,x) = \left\{ v : \lim_{y\to x, y\in C, \theta\to 0_+} \frac{d(y+\theta v,C)}{\theta} = 0 \right\} $$
• A set is regular if $T(x,C) = T_C(x,C)$ for all $x\in C$.

My questions are a bit general, as I'm trying to build intuition.

1. If a set is convex, then is it always regular? (Including possibly infinitely-dimensional sets? What if we restrict to finite-dimensional sets?) Would it be fair to say that here, both definitions boil down to the "usual" tangent cone definition, e.g. $$ T_0(C,x) = \lim_{r\to 0}\mathrm{cone}(\{y\in C: \|x-y\|\leq r\}) $$
2. Do funny things happen if $C$ is a low dimensional subspace (e.g. convex but unbounded and with empty interior?)
3. Now assume that I have a set which is nonconvex, shaped like a cashew (e.g. no nonsmooth points.) Then it seems like the tangent cone at any point is just a halfspace, using either definition. Does this seem true?
4. Now assume that I have a set which is "pointy" and nonconvex, like Pacman. In particular, take $x$ to be the point most inside Pacman's mouth. More precisely, consider $$ C = \{x : \|x\| \leq 1\} \cap \{x : \angle(x_2,x_1) > \alpha \text{ or }\angle (x_2,x_1) < \alpha\} $$for some $\pi/2 > \alpha > 0$, and take $x = 0$. I suppose the tangent cone, using either definition, at this point, is the set $\{x : \angle(x_2,x_1) > \alpha \text{ or }\angle (x_2,x_1) < \alpha\}$, and the normal cone, defined as the polar of the tangent cone, is empty (in both definitions). Does this sound sensible?
5. Finally, the main question is: what is an example of a set which is not regular? I presume such sets must be nonconvex; can they also be finite-dimensional? What about compact / closed / bounded?

Thanks for any discussion!

Answer 1

I answer the first question.

"If $C$ is convex, Clarke tangent cone is the closed tangent cone in convex analysis"- cited https://sites.math.washington.edu/~rtr/papers/rtr078-ClarkeTanCone.pdf

If $C$ is Star shaped at $x$ (a generalized form of convex), Bouligand tangent cone of $C$ at $x$ is also the closed tangent cone in convex analysis, you can see this result in Corollary 4.11, Johannes Jahn - Introduction to the theory of nonlinear optimization-Springer (2007). So, if $C$ is convex, we have that $C$ is regular and three types of tangent cones mentioned above are identical.

You can also find some counterexample for not regular cone in the article above.

Sorry but your questions are really hard and the answers require much effort and time to be demonstrated in detail. Therefore I just can give you some documents including useful results. Hope they can help you.

Additionally, this is a topic that I am concerning, so you can contact me for a discussion about this. My email is [email protected]

Best wishes.